Algebra and geometry in several complex variables by Palamodov V.

By Palamodov V.

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Q the analytic set Zj = {Pj = 0} is irreducible, (iii) the set Sing (Zj ) is analytic set of dimension < n − 1 and Zj \ Sing (Zj ) is a connected analytic manifold of dimension n − 1 which is dense in Zj . Proof. Assume, first, that k = 1. We call ∆ = {S1 (P ) = 0, z ∈ B} the discriminant set of P. The complement B\∆ is connected. Note that the set Z\P −1 (∆) is an analytic manifold of dimension n − 1. Fix a point b ∈ B\∆ and consider the fundamental group π1 (B\∆) of loops through b. 2). , αm (z ) along this loop.

Mk (B) = mk+1 (B) . By Nakayama’s lemma we conclude that mk (B) = 0 which implies the inclusion mk (A) ⊂ I. This completes the proof. Problem 4. Show that always mk (A) ⊂ I holds for k = dimC A/I. , zn ] and algebraic varieties Z ⊂ Cn . Theorem 11 If the set Z of common roots of elements of an ideal I is empty, then I contains the unit element. Proof. , ps of I. They have no common root. , gs such that g1 p1 + 6 ... + gs ps = 1. We use induction in n. For n = 0 the statement is trivial. For n > 0 we choose a coordinate system in such a way that all the highest power of zn in pj has constant coefficient for all j.

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